Fractional double-phase patterns: concentration and multiplicity of solutions

This paper deals with the study of the following nonlinear doubly nonlocal equation: (−Δ)ps1u + β(−Δ)qs2u = λa(x)|u|δ−2u + b(x)|u|r−2u,in Ω,u = 0in ℝn∖Ω, where Ω is a bounded domain in ℝn with smoo...

Regularity and multiplicity results for fractional (p,q)-Laplacian equations

Multiplicity and Concentration of Positive Solutions for Fractional Unbalanced Double-Phase Problems

Abstract The aim of this paper is to study the existence and multiplicity of solutions for a class of fractional Kirchho problems involving Choquard type nonlinearity and singular nonlinearity. Under suitable assumptions, two nonnegative and nontrivial solutions are obtained by using the Nehari manifold approach combined with the Hardy-Littlehood-Sobolev inequality.

Combined effects of Choquard and singular nonlinearities in fractional Kirchhoff problems

This article deals with a survey of recent developments and results on Choquard equations where we focus on the existence and multiplicity of solutions of the partial differential equations which involves the nonlinearity of the convolution type. Because of its nature, these equations are categorized under the nonlocal problems. We give a brief survey on the work already done in this regard following which we illustrate the problems we have addressed. Seeking the help of variational methods and asymptotic estimates, we prove our main results.

Critical Growth Elliptic Problems with Choquard Type Nonlinearity: A Survey

We consider the following Kirchhoff–Choquard equation − M ( ‖ ∇ u ‖ L 2 2 ) Δ u = λ f ( x ) | u | q − 2 u + ∫ Ω | u ( y ) | 2 μ ∗ | x − y | μ d y | u | 2 μ ∗ − 2 u in Ω , u = 0 on ∂ Ω , where Ω is a bounded domain in R N ( N ≥ 3 ) with C 2 boundary, 2 μ ∗ = 2 N − μ N − 2 , 1 q ≤ 2 , and f is a continuous real valued sign changing function. When 1 q 2 , using the method of Nehari manifold and Concentration-compactness Lemma, we prove the existence and multiplicity of positive solutions of the above problem. We also prove the existence of a positive solution when q = 2 using the Mountain Pass Lemma.

Divya Goel

Papers

Regularity and multiplicity results for fractional (p,q)-Laplacian equations

Kirchhoff equations with Hardy-Littlewood-Sobolev critical nonlinearity

Regularity results on a class of doubly nonlocal problems

Coron Problem for Nonlocal Equations Involving Choquard Nonlinearity

On the second eigenvalue of combination between local and nonlocal -Laplacian